Life at Low D. Lecture I : Biological Filaments (1D)

Transcription

1 ife at ow D ecture I : Biological Filaments (1D)

2 Cytoskeleton The "scaffolding" or "skeleton" contained within the cytoplasm Cytoskeleton comprised of 3 types of protein filaments: a) Actin b) Microtubules c) Intermediate Filaments Endothelial cells. Nuclei are stained blue (DAPI), microtubles green (antibody bound to FITC), and actin filaments are red (phalloidin bound to TRITC). Bovine pulmonary arthery endothelial cells

3 A Drosophila tissue culture cell labeled for: microtubules in green and DNA in blue ab/imagegallery.html Actin filaments of mouse embryo fibroblasts, stained with FITC-phalloidin

4 Mitosis A Chinese hamster ovary cell (CHO) cell in anaphase. Actin (red), tubulin (green) and DNA (blue) are labeled A mitotic spindle with kinetochores (motors that bend microtubules, thereby destabilizing them and promoting depolymerization) in red and microtubules stained green people.virginia.edu/~djb6t/abweb/anne.htm

5 Biological Filaments Actin G-Actin ADP and the divalent cation highlighted ATP ATP-Bound Actin 3kDa ~ 3nm 3 atomic structure of an actin filament with13 subunits, based on actin filamenmodel of Ken Holmes; surface representation, rendered with PyMol

6 Biological Filaments Microtubules Trends in Biotechnology Vol.6 No.6 Tubulin 5nm

7 Biological Filaments from Actin and Microtubules Flagella

8 Biological Filaments Filamentous Bacteriophages (M13) geneiii and genevi proteins infective tip ~ 1µm geneviii protein ~5aa monomer, 7 copies 6nm EurBiophysJ 37. p.51(8) 6nm genevii and geneix proteins remote tip ssdna 759bp 177nm

9 Biological Filaments Phys. Biol. Of the Cell

10 Biological Filaments What are the types of deformations encountered in a filament s lifetime? Stretch Bend Twist

11 Stretching (Stress and Strain) A

12 Stretching (Stress and Strain) A F Stress σ F A

13 Stretching (Stress and Strain) A F A F Stress σ F A Strain ε

14 Stretching (Stress and Strain) A F A F Stress σ F A Strain ε

15 Stretching (Stress and Strain) A F A F Stress σ F A Strain ε E σ Eε Force Area Energy Volume

16 Peruvian Example Rubber Band 1cm.1cm Stress Produce of Peru A. 1cm F Stretched: Strain Relaxed: F 1lbs. 5kg ~ (.5kg)(9.8 m ) 5N σ F A s 5 N cm σ Eε ε.75 E 3 N. 3GPa cm

17 Young s Modulus Stress σ F A Strain ε σ Eε E Young s Modulus UNITS!!! Force Energy E Area Volume Estimates Metals G-Actin 15pm U ~ ev int E GPa 3 V ~ 3nm U int ~ 1k B T E GPa MATERIA E [ G Pa] Diamond 1 Steel 11 Glass 1 Wood 1 Plastic. Rubber. 3 1Pa 1N / m 1J / m

18 Spring Analogy F F A a F kx U 1 kx

19 Spring Analogy F F A a + δa F k( δa) U 1 k( δa)

20 Spring Analogy F F + A a + δa F A EA E F ( ) F k( δa) U 1 EA ( ) U 1 k( δa)

21 Strain Energy + A F F A E U Strain 1 EA ( )

22 Strain Energy E A F ( ) 1 EA U Strain A F + V E A E U Strain 1 1 ε

23 Strain Energy + A F F A E U Strain 1 EA ( ) 1 1 U Strain E A Eε V U Strain Volume 1 Eε E U Strain ε ( x, y, z) dxdydz

24 Bending a

25 Bending a + δa a δa

26 Bending z x Rθ θ R (z) ( ) z ( z) ( z) ( R + z) R R ε ( z) z R

27 R θ z R z R z R z z ) ( ) ( ) ( + ) ( (z) x Rθ R z z ) ( ε dxdydz z y x E U Strain ),, ( ε Bending

28 R θ z x Rθ R z z ) ( ε dxdydz z y x E U Strain ),, ( ε h h 3 / / / / R h E h h R E dz R z dy dx E U h h h h Bend Bending

29 Bend Energy U Bend E h 1 R h h

30 Bend Energy U Bend E h 1 R h h U Bend E π r R r

31 Bend Energy U Bend U Bend E h 1 R E π r R h h r Areal Moment of Inertia I 3 wh 1 I π r I ( r π r )

32 Bend Energy U Bend EI R R Areal Moment of Inertia I 3 wh 1 I π r I ( r π r )

33 Bend Energy U Bend EI R R Areal Moment of Inertia I 3 wh 1 I π r I ( r π r )

34 Bend Energy U Bend EI R Energy required to bend filament into circular arc of ength,, and radius, R R πr U oop YI(πR) R πei R

35 Bend Energy U Bend EI R Energy required to bend filament into circular arc of Radius, R, and ength R R R U YI( R) EI EI Rad R R

36 Bend Energy U Bend EI R Energy required to bend filament into circular arc of Radius, R, and ength R R R 1 k EI B T ξ U YI( R) EI EI Rad R R ξ EI k T B

37 Persistence ength ξ EI k T B Correlation of Tangent Angles Oooooh what fun! Just wait until ecture II

38 Persistence ength DNA ξ EI k T B

39 Persistence ength DNA ξ EI k T B I R π.78nm ξ 3nm E GPa

40 Persistence ength DNA ξ EI k T B Smith, et.al Science 71 (1996) p.795 F σ A πr σ Eε F ε ο 1 Α r 8,5bp 16.µ m F E. 3GPa π r

41 Persistence ength DNA ξ EI k T B I R π.78nm 66nm ξ E. 3GPa

42 Persistence ength Actin ξ EI k T B

43 Persistence ength Actin ξ EI k T B R nm I R π.78nm ξ 7µm E 1. 6GPa

44 Persistence ength Microtubules ξ EI k T B

45 Persistence ength Microtubules ξ EI k T B R 5nm 1. R 1nm I ( R R ) π.78nm E 1. 6GPa.5mm ξ

46 Persistence ength A length dependence? U Strain E dxdydz ε ε ( x, y, z) PNAS (6)

47 Stretching Experiments M13 virus nm Polystyrene Bead Belcher, A. et al. PNAS 1 (7) p.89

48 Stretching Experiments Actin Tsuda et al. PNAS 93 (1996)

49 ife at ow D END ecture I Thank you.

50 ife at ow D ecture II: Vitrual ab

51 Bend Energy U Bend EI R Energy required to bend filament of ength into circular arc of Radius, R R

52 Bend Energy U Bend EI R Energy required to bend filament of ength into circular arc of Radius, R R s s U Bend U o Bend all arcs EI R 1 ds ( s)

53 Bend Energy U Bend EI R 1 ds ( s) s s R dθ ds ds Rdθ U Bend EI 1 R dθ ds d ds θ ds

54 Bend Energy U Bend EI d ds θ ds

55 Molecular Model - Estimates from Molecular Interactions F k ( a) S δ a + δa Monomer Monomer G-Actin ε δa F k a σ Sδ σ ε a a a K (N/m) a a (A) E k S k S a actin ks N m actin a ο 1Α

56 Bending Deformations (as collection of stretched beams) i YI j YI E Bend E Bend R R j i YI EBend R n n E Bend YI R 1 ds ( s)

57 Fix one end Displace rod by amount x Bending Moment x / R Flexural Rigidity E Bx YI E Bend 3 B R R Harmonic Oscillator k E x where k B 3 x k B k T 3 k B B T Compare deflections for the filament models

58 Energies and engthscales 1 1k B T 1 J Gigajoule - 1 billion joules. 1J 1N Six gigajoules is about the amount of chemical energy in a barrel of oil. Terajoule - 1 trillion joules. m About 6 terajoules were released by the bomb that exploded over Hiroshima. 1 k B T pn nm

59 Molecular Model F k ( a) S δ a + δa δa F ε σ a a σ k S a ε E k S a Monomer Monomer K (N/m) a (A) actin ks N m actin a ο 1Α G-Actin